Thomas Tautenhahn

Scheduling a batching machine

textabstractWe study the problem of scheduling a chain-reentrant shop, in which each job goes for its processing first to a machine called the primary machine, then to a number of other machines in a fixed sequence, and finally back to the primary machine for its last operation. The problem is to schedule the jobs so as to minimize the makespan. This problem is unary NP-hard for a general number of machines. We focus in particular on the two-machine case that is also at least binary NP-hard. We prove some properties that identify a specific class of optimal schedules, and then use these properties in designing an approximation algorithm and a branch-and-bound type optimization algorithm. The approximation algorithm, of which we present three versions, has a worst-case performance guarantee of f32 along with an excellent empirical performance. The optimization algorithm solves large instances quickly. Finally, we identify a few well solvable special cases and present a pseudo-polynomial algorithm for the case in which the first and the last operations of any job (on the primary machine) are identical.

Approximability and nonapproximability results for minimizing total flow time on a single machine

We consider the problem of scheduling n jobs that are released over time on a single machine in order to minimize the total ??flow time??. This problem is well-??known to be NP??-complete, and the best polynomial time approximation algorithms constructed so far had (more or less trivial)?? worst-??case performance guarantees of O??(n).???? In this paper, we present one positive and one negative result on polynomial time approximations for the minimum total ??flow time problem??. The positive result is the first approxima??tion algorithm with a sublinear worst-??case performance guarantee of O(\sqrt{n}). This algorithm is based on resolving the preemptions of the corresponding optimum preemptive schedule??. The performance guarantee of our approximation algorithm is not far from best possible as our second, negative result demonstrates.?? Unless P=NP, no polynomial time approxima??tion algorithm for minimum total ??flow time can have a worst-??case performance guarantee of O(n^{1/2 - \epsilon}) for any \epsilon > 0. ?? ?? ???? Keywords:?? scheduling, approximation algorithm, worst-??case analysis, total flow time, release time, single machine??.

Constructive heuristic algorithms for the open shop problem

In this paper we consider constructive heuristic algorithms for the open shop problem with minimization of the schedule length. By means of investigations of the structure of a feasible solution two types of heuristic algorithms are developed: construction of a rank-minimal schedule by solving successively weighted maximum cardinality matching problems and construction of an approximate schedule by applying insertion techniques combined with beam search. All presented algorithms are tested on benchmark problems from the literature. Our computational results demonstrate the excellent solution quality of our insertion algorithm, especially for greater job and machine numbers. For 29 of 30 benchmark problems with at least 10 jobs and 10 machines we improve the best known values obtained by tabu search.ZusammenfassungMit dem Ziel der Minimierung der Gesamtbearbeitungszeit werden konstruktive Heuristiken für das open shop Problem betrachtet. Durch strukturelle Untersuchungen einer zulässigen Lösung werden zwei Arten von Heuristiken entwickelt: Konstruktion eines rangminimalen Bearbeitungsplanes durch sukzessives Lösen von gewichteten Matchingproblemen mit maximaler Kardinalität und Konstruktion einer Näherungslösung durch Anwendung von Einfügungstechniken kombiniert mit beam search. Die Verfahren werden an den aus der Literatur bekannten Benchmark Beispielen getestet. Die Resultate unserer Testrechnungen demonstrieren eindrucksvoll die Qualität unseres Einfügungsalgorithmus, insbesondere für wachsende Auftrags- und Maschinenzahl. Für 29 der 30 Benchmark Beispiele mit mindestens 10 Aufträgen und 10 Maschinen wird die mit Tabusuche ermittelte Näherungslösung verbessert.

Scheduling batches with simultaneous job processing for two‐machine shop problems

We consider the problem of scheduling independent jobs on two machines in an open shop, a job shop and a flow shop environment. Both machines are batching machines, which means that several operations can be combined into a batch and processed simultaneously on a machine. The batch processing time is the maximum processing time of operations in the batch, and all operations in a batch complete at the same time. Such a situation may occur, for instance, during the final testing stage of circuit board manufacturing, where burn-in operations are performed in ovens. We consider cases in which there is no restriction on the size of a batch on a machine, and in which a machine can process only a bounded number of operations in one batch. For most of the possible combinations of restrictions, we establish the complexity status of the problem.

A comparison of heuristic algorithms for flow shop scheduling problems with setup times and limited batch size

In this paper, we propose different heuristic algorithms for flow shop scheduling problems, where the jobs are partitioned into groups or families. Jobs of the same group can be processed together in a batch but the maximal number of jobs in a batch is limited. A setup is necessary before starting the processing of a batch, where the setup time depends on the group of the jobs. In this paper, we consider the case when the processing time of a batch is given by the maximum of the processing times of the operations contained in the batch. As objective function we consider the makespan as well as the weighted sum of completion times of the jobs. For these problems, we propose and compare various constructive and iterative algorithms. We derive suitable neighbourhood structures for such problems with batch setup times and describe iterative algorithms that are based on different types of local search algorithms. Except for standard metaheuristics, we also apply multilevel procedures which use different neighbourhoods within the search. The algorithms developed have been tested in detail on a large collection of problems with up to 120 jobs.

Approximability and Nonapproximability Results for Minimizing Total Flow Time on a Single Machine

We consider the problem of scheduling n jobs that are released over time on a single machine in order to minimize the total flow time. This problem is well known to be NP-complete, and the best polynomial-time approximation algorithms constructed so far had (more or less trivial) worst-case performance guarantees of O(n). In this paper, we present one positive and one negative result on polynomial-time approximations for the minimum total flow time problem: The positive result is the first approximation algorithm with a sublinear worst-case performance guarantee of

Heuristic constructive algorithms for open shop scheduling to minimize mean flow time

O(\sqrt{n})

Scheduling multi‐operation jobs on a single machine

. This algorithm is based on resolving the preemptions of the corresponding optimum preemptive schedule. The performance guarantee of our approximation algorithm is not far from best possible, as our second, negative result demonstrates: Unless P=NP, no polynomial-time approximation algorithm for minimum total flow time can have a worst-case performance guarantee of

Heuristics for permutation flow shop scheduling with batch setup times

O(n^{1/2-\eps})

Scheduling unit time open shops to minimize the weighted number of late jobs

for any